c.
ĐKXĐ: \(sinx\ne0\Rightarrow x\ne k\pi\)
\(1-\dfrac{\sqrt{3}cosx}{sinx}-4cosx=0\)
\(\Rightarrow sinx-\sqrt{3}cosx-4sinx.cosx=0\)
\(\Leftrightarrow sinx-\sqrt{3}cosx=2sin2x\)
\(\Leftrightarrow\dfrac{1}{2}sinx-\dfrac{\sqrt{3}}{2}cosx=sin2x\)
\(\Leftrightarrow sin2x=sin\left(x-\dfrac{\pi}{3}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}2x=x-\dfrac{\pi}{3}+k2\pi\\2x=\dfrac{4\pi}{3}-x+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{3}+k2\pi\\x=\dfrac{4\pi}{9}+\dfrac{k2\pi}{3}\end{matrix}\right.\)
a.
\(\Leftrightarrow3sin3x-4sin^34x-\sqrt{3}cos9x=2sin2x\)
\(\Leftrightarrow sin9x-\sqrt{3}cos9x=2sin2x\)
\(\Leftrightarrow\dfrac{1}{2}sin9x-\dfrac{\sqrt{3}}{2}cos9x=sin2x\)
\(\Leftrightarrow sin\left(9x-\dfrac{\pi}{3}\right)=sin2x\)
\(\Leftrightarrow\left[{}\begin{matrix}9x-\dfrac{\pi}{3}=2x+k2\pi\\9x-\dfrac{\pi}{3}=\pi-2x+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{21}+\dfrac{k2\pi}{7}\\x=\dfrac{4\pi}{33}+\dfrac{k2\pi}{11}\end{matrix}\right.\)
b.
\(\Leftrightarrow\sqrt{3}cosx+sin3x-sinx-sin3x=\sqrt{2}\)
\(\Leftrightarrow\sqrt{3}cosx-sinx=\sqrt{2}\)
\(\Leftrightarrow\dfrac{\sqrt{3}}{2}cosx-\dfrac{1}{2}sinx=\dfrac{\sqrt{2}}{2}\)
\(\Leftrightarrow cos\left(x+\dfrac{\pi}{6}\right)=cos\left(\dfrac{\pi}{4}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}x+\dfrac{\pi}{6}=\dfrac{\pi}{4}+k2\pi\\x+\dfrac{\pi}{6}=-\dfrac{\pi}{4}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{12}+k2\pi\\x=-\dfrac{5\pi}{12}+k2\pi\end{matrix}\right.\)
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